91Ó°ÊÓ

Find an equation of the line \(l\) tangent to the graph of \(f\) at the given point. \(f(x)=-2 \cos 3 x ;(\pi / 3,2)\)

Suppose \(f\) and \(g\) are functions such that \(f(a)=g(a)\), and assume that \(f^{\prime}(a)\) and \(g^{\prime}(a)\) exist. Does it follow that \(f^{\prime}(a)\) \(=g^{\prime}(a) ?\) Explain.

Suppose that \(f^{\prime}(a)\) and \(g^{\prime}(a)\) exist, and that \(f(a) \neq 0\) and \(g(a) \neq 0\) a. Let \(h=f g .\) Show that \(\frac{h^{\prime}(a)}{h(a)}=\frac{f^{\prime}(a)}{f(a)}+\frac{g^{\prime}(a)}{g(a)}\). b. Let \(k=f / g .\) Show that \(\frac{k^{\prime}(a)}{k(a)}=\frac{f^{\prime}(a)}{f(a)}-\frac{g^{\prime}(a)}{g(a)}\).

Suppose \(y=8 x-23\) is an equation of the line tangent to the graph of a function \(f\) at \((5,17)\). Find \(f^{\prime}(5)\).

Find a formula for the \(n\) th derivative of \(f\), for \(n \geq 1 .\) $$ f(x)=1 / x $$

Find an equation of the line \(l\) tangent to the graph of \(f\) at the given point. \(f(x)=2 e^{-3 x} ;(0,2)\)

Suppose \(f\) is differentiable at \(a\), and let $$ g(x)=\left\\{\begin{array}{cl} \frac{f(x)-f(a)}{x-a} & \text { if } x \neq a \\ f^{\prime}(a) & \text { if } x=a \end{array}\right. $$

Find a formula for the \(n\) th derivative of \(f\), for \(n \geq 1 .\) $$ f(x)=1 /(1-x) $$

Find an equation of the line \(l\) tangent to the graph of \(f\) at the given point. \(f(x)=\ln (\sin x) ;\left(\frac{\pi}{6},-\ln 2\right)\)

Two balls are thrown at the same time, from the same height and in the vertical direction, and with the same initial speed of 10 meters per second. The first ball is thrown upward and the second is thrown downward. Compare the velocities of the two balls.